{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction to Quantitative Finance\n",
    "\n",
    "Copyright (c) 2019 Python Charmers Pty Ltd, Australia, <https://pythoncharmers.com>. All rights reserved.\n",
    "\n",
    "<img src=\"img/python_charmers_logo.png\" width=\"300\" alt=\"Python Charmers Logo\">\n",
    "\n",
    "Published under the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) license. See `LICENSE.md` for details.\n",
    "\n",
    "Sponsored by Tibra Global Services, <https://tibra.com>\n",
    "\n",
    "<img src=\"img/tibra_logo.png\" width=\"300\" alt=\"Tibra Logo\">\n",
    "\n",
    "\n",
    "## Module 1.1: Distributions and Random Processes\n",
    "\n",
    "### 1.1.2: Distributions\n",
    "\n",
    "Many real world phenomena can be modelled with random variables, which is particularly useful when we do not have all possible information about a given environment. Consider a die roll, where we throw a die into the air, and see which number is on top.\n",
    "\n",
    "The die has six possible outcomes: 1, 2, 3, 4, 5 and 6, which are the possible states or values. It is impossible* to get a value other than these six distinct possible.\n",
    "\n",
    "<div class=\"alert alert-warning\">\n",
    "    * Some might argue that it would be possible to properly model a die throw and know what the result would be. Whether it is possible is the type of question that keeps physicists up at night, however it almost certainly would be more effort than it's worth.\n",
    "</div>\n",
    "\n",
    "Here is a simulation of a die roll."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import random\n",
    "\n",
    "possible_states = [1, 2, 3, 4, 5, 6]\n",
    "\n",
    "random.choice(possible_states)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we run this simulation 100,000 times, we can get an *empirical estimate* of the distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "NUM_TRIALS = 100000\n",
    "\n",
    "results = [random.choice(possible_states) for i in range(NUM_TRIALS)]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({1: 16752, 6: 16755, 2: 16819, 5: 16444, 3: 16653, 4: 16577})"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from collections import Counter\n",
    "counts = Counter(results)\n",
    "counts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "RendererRegistry.enable('default')"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import altair as alt\n",
    "import vega\n",
    "alt.renderers.enable('default')  # Setup your environment to show altair plots.  Note this option depends on where you are running your code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   state  count\n",
      "0      1  16752\n",
      "1      2  16819\n",
      "2      3  16653\n",
      "3      4  16577\n",
      "4      5  16444\n",
      "5      6  16755\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "\n",
       "<style>\n",
       "  #altair-viz-d106aa678e3e49cebe1663351ad662be.vega-embed {\n",
       "    width: 100%;\n",
       "    display: flex;\n",
       "  }\n",
       "\n",
       "  #altair-viz-d106aa678e3e49cebe1663351ad662be.vega-embed details,\n",
       "  #altair-viz-d106aa678e3e49cebe1663351ad662be.vega-embed details summary {\n",
       "    position: relative;\n",
       "  }\n",
       "</style>\n",
       "<div id=\"altair-viz-d106aa678e3e49cebe1663351ad662be\"></div>\n",
       "<script type=\"text/javascript\">\n",
       "  var VEGA_DEBUG = (typeof VEGA_DEBUG == \"undefined\") ? {} : VEGA_DEBUG;\n",
       "  (function(spec, embedOpt){\n",
       "    let outputDiv = document.currentScript.previousElementSibling;\n",
       "    if (outputDiv.id !== \"altair-viz-d106aa678e3e49cebe1663351ad662be\") {\n",
       "      outputDiv = document.getElementById(\"altair-viz-d106aa678e3e49cebe1663351ad662be\");\n",
       "    }\n",
       "    const paths = {\n",
       "      \"vega\": \"https://cdn.jsdelivr.net/npm/vega@5?noext\",\n",
       "      \"vega-lib\": \"https://cdn.jsdelivr.net/npm/vega-lib?noext\",\n",
       "      \"vega-lite\": \"https://cdn.jsdelivr.net/npm/vega-lite@5.16.3?noext\",\n",
       "      \"vega-embed\": \"https://cdn.jsdelivr.net/npm/vega-embed@6?noext\",\n",
       "    };\n",
       "\n",
       "    function maybeLoadScript(lib, version) {\n",
       "      var key = `${lib.replace(\"-\", \"\")}_version`;\n",
       "      return (VEGA_DEBUG[key] == version) ?\n",
       "        Promise.resolve(paths[lib]) :\n",
       "        new Promise(function(resolve, reject) {\n",
       "          var s = document.createElement('script');\n",
       "          document.getElementsByTagName(\"head\")[0].appendChild(s);\n",
       "          s.async = true;\n",
       "          s.onload = () => {\n",
       "            VEGA_DEBUG[key] = version;\n",
       "            return resolve(paths[lib]);\n",
       "          };\n",
       "          s.onerror = () => reject(`Error loading script: ${paths[lib]}`);\n",
       "          s.src = paths[lib];\n",
       "        });\n",
       "    }\n",
       "\n",
       "    function showError(err) {\n",
       "      outputDiv.innerHTML = `<div class=\"error\" style=\"color:red;\">${err}</div>`;\n",
       "      throw err;\n",
       "    }\n",
       "\n",
       "    function displayChart(vegaEmbed) {\n",
       "      vegaEmbed(outputDiv, spec, embedOpt)\n",
       "        .catch(err => showError(`Javascript Error: ${err.message}<br>This usually means there's a typo in your chart specification. See the javascript console for the full traceback.`));\n",
       "    }\n",
       "\n",
       "    if(typeof define === \"function\" && define.amd) {\n",
       "      requirejs.config({paths});\n",
       "      require([\"vega-embed\"], displayChart, err => showError(`Error loading script: ${err.message}`));\n",
       "    } else {\n",
       "      maybeLoadScript(\"vega\", \"5\")\n",
       "        .then(() => maybeLoadScript(\"vega-lite\", \"5.16.3\"))\n",
       "        .then(() => maybeLoadScript(\"vega-embed\", \"6\"))\n",
       "        .catch(showError)\n",
       "        .then(() => displayChart(vegaEmbed));\n",
       "    }\n",
       "  })({\"config\": {\"view\": {\"continuousWidth\": 300, \"continuousHeight\": 300}}, \"data\": {\"name\": \"data-6c014555b2ed00ff58bfff3019c115a1\"}, \"mark\": {\"type\": \"bar\"}, \"encoding\": {\"x\": {\"field\": \"state\", \"type\": \"ordinal\"}, \"y\": {\"field\": \"count\", \"type\": \"quantitative\"}}, \"$schema\": \"https://vega.github.io/schema/vega-lite/v5.16.3.json\", \"datasets\": {\"data-6c014555b2ed00ff58bfff3019c115a1\": [{\"state\": 1, \"count\": 16752}, {\"state\": 2, \"count\": 16819}, {\"state\": 3, \"count\": 16653}, {\"state\": 4, \"count\": 16577}, {\"state\": 5, \"count\": 16444}, {\"state\": 6, \"count\": 16755}]}}, {\"mode\": \"vega-lite\"});\n",
       "</script>"
      ],
      "text/plain": [
       "alt.Chart(...)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "\n",
    "die_rolls = pd.DataFrame({\n",
    "    'state': possible_states,\n",
    "    'count': [counts.get(state, 0) for state in possible_states]\n",
    "})\n",
    "\n",
    "print(die_rolls)\n",
    "\n",
    "alt.Chart(die_rolls).mark_bar().encode(\n",
    "    x='state:O',\n",
    "    y='count'\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As one might expect with a \"fair\" die roll like this, the estimates are about the same (your results may vary).\n",
    "\n",
    "More formally, we have a random variable $X$ such that the value of $X$ is the result of rolling a fair die. The distribution of $X$ is the following:\n",
    "\n",
    "\n",
    "$\n",
    "\\begin{gather*}\n",
    "  P(X=x) = \\begin{cases}\n",
    "  \\frac{1}{6} & \\text{if $x \\in \\{1,2,3,4,5,6\\}$}\\\\\n",
    "  0 & \\text{otherwise.}\n",
    "  \\end{cases}\n",
    "\\end{gather*}\n",
    "$\n",
    "\n",
    "In this case, all options (at least, the valid ones) have the same likelihood of appearing, and therefore the resulting distribution is known as a *uniform distribution*. Uniform distributions can occur for both discrete and continuous variables.\n",
    "\n",
    "A **discrete** variable is one that takes on a fixed set of values, such as a die roll or the month you were born. They are usually finite (i.e. there is only so many of them), but the requirement is technically that they be *countable*, which allows for infinite discrete values (think of the integers as a discrete infinite random variable).\n",
    "\n",
    "A **continuous** variable is one that can take on an arbitrary value. As an example, the amount of liquid in a cup is continuous. It can be 350ml, or 350.1ml, or 350.1252342ml, or so on, with infinite precision (we will ignore the effect of the Planck constant). \n",
    "\n",
    "Distributions exist for both discrete random variables, as we saw above, and for continuous random variables, however they often behave differently."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<!-- Run me, but there is  need to worry about this code - \n",
       "    it makes the equation in the next cell larger and more readable -->\n",
       "<style>\n",
       "    .big_function {font-size: 200%;}\n",
       "</style>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%html\n",
    "<!-- Run me, but there is  need to worry about this code - \n",
    "    it makes the equation in the next cell larger and more readable -->\n",
    "<style>\n",
    "    .big_function {font-size: 200%;}\n",
    "</style>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Normal Distributions\n",
    "Let's create a normal distribution, which is a continuous distribution centred around 0 and with a standard deviation of 1.\n",
    "\n",
    "A normal distribution is the most commonly seen continuous distribution, and the one most people are familiar with. The equation for the graph is:\n",
    "\n",
    "$$y = \\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Let's plot that out\n",
    "import numpy as np\n",
    "x = np.linspace(-5, 5, 1000)  # 1000 linearly spaced points starting -5 going to 5\n",
    "\n",
    "mean = 0\n",
    "sigma = 1\n",
    "\n",
    "y = (1 / np.sqrt(2 * np.pi * sigma ** 2)) * np.e ** -((x - mean) ** 2) / (2 * sigma ** 2)\n",
    "\n",
    "# Or a bit more nicely laid out...\n",
    "\n",
    "scale_term = (1 / np.sqrt(2 * np.pi * sigma ** 2))\n",
    "exponent_term = ((x - mean) ** 2) / (2 * sigma ** 2)\n",
    "y = scale_term * np.e ** -exponent_term"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from matplotlib import pyplot as plt\n",
    "plt.plot(x, y, \"-\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = stats.norm(0, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.66040796,  1.6221566 ,  0.78103376, -0.03896793, -0.3957174 ,\n",
       "       -0.46236667,  0.84517539, -1.7958585 ,  1.48547635, -0.66616848])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Generate 10 random variates from this distribution\n",
    "n.rvs(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# note that Counter isn't very helpful here. It is *highly* likely that all counts will be 1\n",
    "# regardless of how many values you generate. In fact, it's nearly impossible to get the same value twice.\n",
    "results = n.rvs(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({0.49595807013821813: 1,\n",
       "         0.5117196913920876: 1,\n",
       "         -0.1187083785680289: 1,\n",
       "         -0.42015424355188685: 1,\n",
       "         0.17241512275781976: 1,\n",
       "         0.8689004192136313: 1,\n",
       "         0.8474101187732266: 1,\n",
       "         0.8250639756474332: 1,\n",
       "         -0.9238044728907774: 1,\n",
       "         -0.39901959000095666: 1})"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Counter(results)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<style>\n",
       "  #altair-viz-9e4bce566532455fbe891b0998871e1d.vega-embed {\n",
       "    width: 100%;\n",
       "    display: flex;\n",
       "  }\n",
       "\n",
       "  #altair-viz-9e4bce566532455fbe891b0998871e1d.vega-embed details,\n",
       "  #altair-viz-9e4bce566532455fbe891b0998871e1d.vega-embed details summary {\n",
       "    position: relative;\n",
       "  }\n",
       "</style>\n",
       "<div id=\"altair-viz-9e4bce566532455fbe891b0998871e1d\"></div>\n",
       "<script type=\"text/javascript\">\n",
       "  var VEGA_DEBUG = (typeof VEGA_DEBUG == \"undefined\") ? {} : VEGA_DEBUG;\n",
       "  (function(spec, embedOpt){\n",
       "    let outputDiv = document.currentScript.previousElementSibling;\n",
       "    if (outputDiv.id !== \"altair-viz-9e4bce566532455fbe891b0998871e1d\") {\n",
       "      outputDiv = document.getElementById(\"altair-viz-9e4bce566532455fbe891b0998871e1d\");\n",
       "    }\n",
       "    const paths = {\n",
       "      \"vega\": \"https://cdn.jsdelivr.net/npm/vega@5?noext\",\n",
       "      \"vega-lib\": \"https://cdn.jsdelivr.net/npm/vega-lib?noext\",\n",
       "      \"vega-lite\": \"https://cdn.jsdelivr.net/npm/vega-lite@5.16.3?noext\",\n",
       "      \"vega-embed\": \"https://cdn.jsdelivr.net/npm/vega-embed@6?noext\",\n",
       "    };\n",
       "\n",
       "    function maybeLoadScript(lib, version) {\n",
       "      var key = `${lib.replace(\"-\", \"\")}_version`;\n",
       "      return (VEGA_DEBUG[key] == version) ?\n",
       "        Promise.resolve(paths[lib]) :\n",
       "        new Promise(function(resolve, reject) {\n",
       "          var s = document.createElement('script');\n",
       "          document.getElementsByTagName(\"head\")[0].appendChild(s);\n",
       "          s.async = true;\n",
       "          s.onload = () => {\n",
       "            VEGA_DEBUG[key] = version;\n",
       "            return resolve(paths[lib]);\n",
       "          };\n",
       "          s.onerror = () => reject(`Error loading script: ${paths[lib]}`);\n",
       "          s.src = paths[lib];\n",
       "        });\n",
       "    }\n",
       "\n",
       "    function showError(err) {\n",
       "      outputDiv.innerHTML = `<div class=\"error\" style=\"color:red;\">${err}</div>`;\n",
       "      throw err;\n",
       "    }\n",
       "\n",
       "    function displayChart(vegaEmbed) {\n",
       "      vegaEmbed(outputDiv, spec, embedOpt)\n",
       "        .catch(err => showError(`Javascript Error: ${err.message}<br>This usually means there's a typo in your chart specification. See the javascript console for the full traceback.`));\n",
       "    }\n",
       "\n",
       "    if(typeof define === \"function\" && define.amd) {\n",
       "      requirejs.config({paths});\n",
       "      require([\"vega-embed\"], displayChart, err => showError(`Error loading script: ${err.message}`));\n",
       "    } else {\n",
       "      maybeLoadScript(\"vega\", \"5\")\n",
       "        .then(() => maybeLoadScript(\"vega-lite\", \"5.16.3\"))\n",
       "        .then(() => maybeLoadScript(\"vega-embed\", \"6\"))\n",
       "        .catch(showError)\n",
       "        .then(() => displayChart(vegaEmbed));\n",
       "    }\n",
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{\"value\": -2.5011273157840868}, {\"value\": -0.07224020075681277}, {\"value\": -0.5805665457847065}, {\"value\": -1.6340809017657094}, {\"value\": -0.27742493080117475}]}}, {\"mode\": \"vega-lite\"});\n",
       "</script>"
      ],
      "text/plain": [
       "alt.Chart(...)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Histograms work better here\n",
    "\n",
    "normal_values = pd.DataFrame({\"value\": n.rvs(5000)})\n",
    "\n",
    "alt.Chart(normal_values).mark_bar().encode(\n",
    "    alt.X(\"value\", bin=alt.Bin(maxbins=100)),\n",
    "    y='count()',\n",
    "    color=alt.value('#287E1E'),  # Aside: Note US spelling. Also try \"hex codes\": https://htmlcolorcodes.com/\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "<div class=\"alert alert-success\">\n",
    "Note that Altair won't let us directly compute a histogram with more than 5000 rows. To do that, check out this example, which saves the data to a local temporary json file, and then displays from that. Without doing this, Altair embeds a *copy* of the data in a graph!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<style>\n",
       "  #altair-viz-0c4d497515614344affd37223e4c331d.vega-embed {\n",
       "    width: 100%;\n",
       "    display: flex;\n",
       "  }\n",
       "\n",
       "  #altair-viz-0c4d497515614344affd37223e4c331d.vega-embed details,\n",
       "  #altair-viz-0c4d497515614344affd37223e4c331d.vega-embed details summary {\n",
       "    position: relative;\n",
       "  }\n",
       "</style>\n",
       "<div id=\"altair-viz-0c4d497515614344affd37223e4c331d\"></div>\n",
       "<script type=\"text/javascript\">\n",
       "  var VEGA_DEBUG = (typeof VEGA_DEBUG == \"undefined\") ? {} : VEGA_DEBUG;\n",
       "  (function(spec, embedOpt){\n",
       "    let outputDiv = document.currentScript.previousElementSibling;\n",
       "    if (outputDiv.id !== \"altair-viz-0c4d497515614344affd37223e4c331d\") {\n",
       "      outputDiv = document.getElementById(\"altair-viz-0c4d497515614344affd37223e4c331d\");\n",
       "    }\n",
       "    const paths = {\n",
       "      \"vega\": \"https://cdn.jsdelivr.net/npm/vega@5?noext\",\n",
       "      \"vega-lib\": \"https://cdn.jsdelivr.net/npm/vega-lib?noext\",\n",
       "      \"vega-lite\": \"https://cdn.jsdelivr.net/npm/vega-lite@5.16.3?noext\",\n",
       "      \"vega-embed\": \"https://cdn.jsdelivr.net/npm/vega-embed@6?noext\",\n",
       "    };\n",
       "\n",
       "    function maybeLoadScript(lib, version) {\n",
       "      var key = `${lib.replace(\"-\", \"\")}_version`;\n",
       "      return (VEGA_DEBUG[key] == version) ?\n",
       "        Promise.resolve(paths[lib]) :\n",
       "        new Promise(function(resolve, reject) {\n",
       "          var s = document.createElement('script');\n",
       "          document.getElementsByTagName(\"head\")[0].appendChild(s);\n",
       "          s.async = true;\n",
       "          s.onload = () => {\n",
       "            VEGA_DEBUG[key] = version;\n",
       "            return resolve(paths[lib]);\n",
       "          };\n",
       "          s.onerror = () => reject(`Error loading script: ${paths[lib]}`);\n",
       "          s.src = paths[lib];\n",
       "        });\n",
       "    }\n",
       "\n",
       "    function showError(err) {\n",
       "      outputDiv.innerHTML = `<div class=\"error\" style=\"color:red;\">${err}</div>`;\n",
       "      throw err;\n",
       "    }\n",
       "\n",
       "    function displayChart(vegaEmbed) {\n",
       "      vegaEmbed(outputDiv, spec, embedOpt)\n",
       "        .catch(err => showError(`Javascript Error: ${err.message}<br>This usually means there's a typo in your chart specification. See the javascript console for the full traceback.`));\n",
       "    }\n",
       "\n",
       "    if(typeof define === \"function\" && define.amd) {\n",
       "      requirejs.config({paths});\n",
       "      require([\"vega-embed\"], displayChart, err => showError(`Error loading script: ${err.message}`));\n",
       "    } else {\n",
       "      maybeLoadScript(\"vega\", \"5\")\n",
       "        .then(() => maybeLoadScript(\"vega-lite\", \"5.16.3\"))\n",
       "        .then(() => maybeLoadScript(\"vega-embed\", \"6\"))\n",
       "        .catch(showError)\n",
       "        .then(() => displayChart(vegaEmbed));\n",
       "    }\n",
       "  })({\"config\": {\"view\": {\"continuousWidth\": 300, \"continuousHeight\": 300}}, \"data\": {\"url\": \"altair-temp-data/-e4e8b551560eb3334c77bcbd2c938deb.json\", \"format\": {\"type\": \"json\"}}, \"mark\": {\"type\": \"bar\"}, \"encoding\": {\"x\": {\"bin\": {\"maxbins\": 100}, \"field\": \"value\", \"type\": \"quantitative\"}, \"y\": {\"aggregate\": \"count\", \"type\": \"quantitative\"}}, \"$schema\": \"https://vega.github.io/schema/vega-lite/v5.16.3.json\"}, {\"mode\": \"vega-lite\"});\n",
       "</script>"
      ],
      "text/plain": [
       "alt.Chart(...)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "if not alt.data_transformers.active == 'json':  # Check json isn't already active\n",
    "    import os\n",
    "    # Make a temp folder to put these json files in - Altair creates a lot of them!\n",
    "    dataset_temp_name = 'altair-temp-data/'\n",
    "    if not os.path.exists(dataset_temp_name):\n",
    "        # if the folder doesn't exist, create it\n",
    "        os.mkdir(dataset_temp_name)\n",
    "    # Tell Altair to temporary save datasets it needs to that folder\n",
    "    alt.data_transformers.enable('json', prefix=dataset_temp_name)\n",
    "\n",
    "normal_values = pd.DataFrame({\"value\": n.rvs(100000)})\n",
    "\n",
    "alt.Chart(normal_values).mark_bar().encode(\n",
    "    alt.X(\"value\", bin=alt.Bin(maxbins=100)),\n",
    "    y='count()',\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice the shape, the famous \"bell curve\" of the normal distribution. It's heavily centred around the mean (0) and the spread of data is indicated by the standard deviation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercises\n",
    "\n",
    "1. Plot a histogram of the standard deviation with the following properties:\n",
    "\n",
    "    1. Mean ($\\mu$) of 1, standard deviation ($\\sigma$) of 7\n",
    "    2. $\\mu=10, \\sigma=1$\n",
    "    3. $\\mu=-10, \\sigma=5$\n",
    "2. Create a python function that accepts two inputs (`mean` and `standard_deviation`) and plots the histogram as per question 1.\n",
    "\n",
    "3. Investigate the documentation of Altair and overlay these plots on top of each other, with different colours, giving a result that looks like this:\n",
    "<img src=\"img/snapshot.png\">"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<style>\n",
       "  #altair-viz-40d0f0cf1bdf46f8ab89b36518488c16.vega-embed {\n",
       "    width: 100%;\n",
       "    display: flex;\n",
       "  }\n",
       "\n",
       "  #altair-viz-40d0f0cf1bdf46f8ab89b36518488c16.vega-embed details,\n",
       "  #altair-viz-40d0f0cf1bdf46f8ab89b36518488c16.vega-embed details summary {\n",
       "    position: relative;\n",
       "  }\n",
       "</style>\n",
       "<div id=\"altair-viz-40d0f0cf1bdf46f8ab89b36518488c16\"></div>\n",
       "<script type=\"text/javascript\">\n",
       "  var VEGA_DEBUG = (typeof VEGA_DEBUG == \"undefined\") ? {} : VEGA_DEBUG;\n",
       "  (function(spec, embedOpt){\n",
       "    let outputDiv = document.currentScript.previousElementSibling;\n",
       "    if (outputDiv.id !== \"altair-viz-40d0f0cf1bdf46f8ab89b36518488c16\") {\n",
       "      outputDiv = document.getElementById(\"altair-viz-40d0f0cf1bdf46f8ab89b36518488c16\");\n",
       "    }\n",
       "    const paths = {\n",
       "      \"vega\": \"https://cdn.jsdelivr.net/npm/vega@5?noext\",\n",
       "      \"vega-lib\": \"https://cdn.jsdelivr.net/npm/vega-lib?noext\",\n",
       "      \"vega-lite\": \"https://cdn.jsdelivr.net/npm/vega-lite@5.16.3?noext\",\n",
       "      \"vega-embed\": \"https://cdn.jsdelivr.net/npm/vega-embed@6?noext\",\n",
       "    };\n",
       "\n",
       "    function maybeLoadScript(lib, version) {\n",
       "      var key = `${lib.replace(\"-\", \"\")}_version`;\n",
       "      return (VEGA_DEBUG[key] == version) ?\n",
       "        Promise.resolve(paths[lib]) :\n",
       "        new Promise(function(resolve, reject) {\n",
       "          var s = document.createElement('script');\n",
       "          document.getElementsByTagName(\"head\")[0].appendChild(s);\n",
       "          s.async = true;\n",
       "          s.onload = () => {\n",
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       "          };\n",
       "          s.onerror = () => reject(`Error loading script: ${paths[lib]}`);\n",
       "          s.src = paths[lib];\n",
       "        });\n",
       "    }\n",
       "\n",
       "    function showError(err) {\n",
       "      outputDiv.innerHTML = `<div class=\"error\" style=\"color:red;\">${err}</div>`;\n",
       "      throw err;\n",
       "    }\n",
       "\n",
       "    function displayChart(vegaEmbed) {\n",
       "      vegaEmbed(outputDiv, spec, embedOpt)\n",
       "        .catch(err => showError(`Javascript Error: ${err.message}<br>This usually means there's a typo in your chart specification. See the javascript console for the full traceback.`));\n",
       "    }\n",
       "\n",
       "    if(typeof define === \"function\" && define.amd) {\n",
       "      requirejs.config({paths});\n",
       "      require([\"vega-embed\"], displayChart, err => showError(`Error loading script: ${err.message}`));\n",
       "    } else {\n",
       "      maybeLoadScript(\"vega\", \"5\")\n",
       "        .then(() => maybeLoadScript(\"vega-lite\", \"5.16.3\"))\n",
       "        .then(() => maybeLoadScript(\"vega-embed\", \"6\"))\n",
       "        .catch(showError)\n",
       "        .then(() => displayChart(vegaEmbed));\n",
       "    }\n",
       "  })({\"config\": {\"view\": {\"continuousWidth\": 300, \"continuousHeight\": 300}}, \"data\": {\"url\": \"altair-temp-data/-1cb17f2eb89525fc84a20fbb90b4aaae.json\", \"format\": {\"type\": \"json\"}}, \"mark\": {\"type\": \"bar\"}, \"encoding\": {\"x\": {\"bin\": {\"maxbins\": 100}, \"field\": \"value\", \"type\": \"quantitative\"}, \"y\": {\"aggregate\": \"count\", \"type\": \"quantitative\"}}, \"title\": \"Normal Distribution histogram with mean 1 and standard deviation 7\", \"$schema\": \"https://vega.github.io/schema/vega-lite/v5.16.3.json\"}, {\"mode\": \"vega-lite\"});\n",
       "</script>"
      ],
      "text/plain": [
       "alt.Chart(...)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "\n",
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       "    width: 100%;\n",
       "    display: flex;\n",
       "  }\n",
       "\n",
       "  #altair-viz-01a9e97e1aaf4e49a7930536dddd0f98.vega-embed details,\n",
       "  #altair-viz-01a9e97e1aaf4e49a7930536dddd0f98.vega-embed details summary {\n",
       "    position: relative;\n",
       "  }\n",
       "</style>\n",
       "<div id=\"altair-viz-01a9e97e1aaf4e49a7930536dddd0f98\"></div>\n",
       "<script type=\"text/javascript\">\n",
       "  var VEGA_DEBUG = (typeof VEGA_DEBUG == \"undefined\") ? {} : VEGA_DEBUG;\n",
       "  (function(spec, embedOpt){\n",
       "    let outputDiv = document.currentScript.previousElementSibling;\n",
       "    if (outputDiv.id !== \"altair-viz-01a9e97e1aaf4e49a7930536dddd0f98\") {\n",
       "      outputDiv = document.getElementById(\"altair-viz-01a9e97e1aaf4e49a7930536dddd0f98\");\n",
       "    }\n",
       "    const paths = {\n",
       "      \"vega\": \"https://cdn.jsdelivr.net/npm/vega@5?noext\",\n",
       "      \"vega-lib\": \"https://cdn.jsdelivr.net/npm/vega-lib?noext\",\n",
       "      \"vega-lite\": \"https://cdn.jsdelivr.net/npm/vega-lite@5.16.3?noext\",\n",
       "      \"vega-embed\": \"https://cdn.jsdelivr.net/npm/vega-embed@6?noext\",\n",
       "    };\n",
       "\n",
       "    function maybeLoadScript(lib, version) {\n",
       "      var key = `${lib.replace(\"-\", \"\")}_version`;\n",
       "      return (VEGA_DEBUG[key] == version) ?\n",
       "        Promise.resolve(paths[lib]) :\n",
       "        new Promise(function(resolve, reject) {\n",
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       "          document.getElementsByTagName(\"head\")[0].appendChild(s);\n",
       "          s.async = true;\n",
       "          s.onload = () => {\n",
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       "          };\n",
       "          s.onerror = () => reject(`Error loading script: ${paths[lib]}`);\n",
       "          s.src = paths[lib];\n",
       "        });\n",
       "    }\n",
       "\n",
       "    function showError(err) {\n",
       "      outputDiv.innerHTML = `<div class=\"error\" style=\"color:red;\">${err}</div>`;\n",
       "      throw err;\n",
       "    }\n",
       "\n",
       "    function displayChart(vegaEmbed) {\n",
       "      vegaEmbed(outputDiv, spec, embedOpt)\n",
       "        .catch(err => showError(`Javascript Error: ${err.message}<br>This usually means there's a typo in your chart specification. See the javascript console for the full traceback.`));\n",
       "    }\n",
       "\n",
       "    if(typeof define === \"function\" && define.amd) {\n",
       "      requirejs.config({paths});\n",
       "      require([\"vega-embed\"], displayChart, err => showError(`Error loading script: ${err.message}`));\n",
       "    } else {\n",
       "      maybeLoadScript(\"vega\", \"5\")\n",
       "        .then(() => maybeLoadScript(\"vega-lite\", \"5.16.3\"))\n",
       "        .then(() => maybeLoadScript(\"vega-embed\", \"6\"))\n",
       "        .catch(showError)\n",
       "        .then(() => displayChart(vegaEmbed));\n",
       "    }\n",
       "  })({\"config\": {\"view\": {\"continuousWidth\": 300, \"continuousHeight\": 300}}, \"data\": {\"url\": \"altair-temp-data/-4138241a11c1f79e076acebbe4cbf64c.json\", \"format\": {\"type\": \"json\"}}, \"mark\": {\"type\": \"bar\"}, \"encoding\": {\"x\": {\"bin\": {\"maxbins\": 100}, \"field\": \"value\", \"type\": \"quantitative\"}, \"y\": {\"aggregate\": \"count\", \"type\": \"quantitative\"}}, \"title\": \"Normal Distribution histogram with mean 10 and standard deviation 1\", \"$schema\": \"https://vega.github.io/schema/vega-lite/v5.16.3.json\"}, {\"mode\": \"vega-lite\"});\n",
       "</script>"
      ],
      "text/plain": [
       "alt.Chart(...)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "\n",
       "<style>\n",
       "  #altair-viz-2c42447db68d48638ed8443a283ade97.vega-embed {\n",
       "    width: 100%;\n",
       "    display: flex;\n",
       "  }\n",
       "\n",
       "  #altair-viz-2c42447db68d48638ed8443a283ade97.vega-embed details,\n",
       "  #altair-viz-2c42447db68d48638ed8443a283ade97.vega-embed details summary {\n",
       "    position: relative;\n",
       "  }\n",
       "</style>\n",
       "<div id=\"altair-viz-2c42447db68d48638ed8443a283ade97\"></div>\n",
       "<script type=\"text/javascript\">\n",
       "  var VEGA_DEBUG = (typeof VEGA_DEBUG == \"undefined\") ? {} : VEGA_DEBUG;\n",
       "  (function(spec, embedOpt){\n",
       "    let outputDiv = document.currentScript.previousElementSibling;\n",
       "    if (outputDiv.id !== \"altair-viz-2c42447db68d48638ed8443a283ade97\") {\n",
       "      outputDiv = document.getElementById(\"altair-viz-2c42447db68d48638ed8443a283ade97\");\n",
       "    }\n",
       "    const paths = {\n",
       "      \"vega\": \"https://cdn.jsdelivr.net/npm/vega@5?noext\",\n",
       "      \"vega-lib\": \"https://cdn.jsdelivr.net/npm/vega-lib?noext\",\n",
       "      \"vega-lite\": \"https://cdn.jsdelivr.net/npm/vega-lite@5.16.3?noext\",\n",
       "      \"vega-embed\": \"https://cdn.jsdelivr.net/npm/vega-embed@6?noext\",\n",
       "    };\n",
       "\n",
       "    function maybeLoadScript(lib, version) {\n",
       "      var key = `${lib.replace(\"-\", \"\")}_version`;\n",
       "      return (VEGA_DEBUG[key] == version) ?\n",
       "        Promise.resolve(paths[lib]) :\n",
       "        new Promise(function(resolve, reject) {\n",
       "          var s = document.createElement('script');\n",
       "          document.getElementsByTagName(\"head\")[0].appendChild(s);\n",
       "          s.async = true;\n",
       "          s.onload = () => {\n",
       "            VEGA_DEBUG[key] = version;\n",
       "            return resolve(paths[lib]);\n",
       "          };\n",
       "          s.onerror = () => reject(`Error loading script: ${paths[lib]}`);\n",
       "          s.src = paths[lib];\n",
       "        });\n",
       "    }\n",
       "\n",
       "    function showError(err) {\n",
       "      outputDiv.innerHTML = `<div class=\"error\" style=\"color:red;\">${err}</div>`;\n",
       "      throw err;\n",
       "    }\n",
       "\n",
       "    function displayChart(vegaEmbed) {\n",
       "      vegaEmbed(outputDiv, spec, embedOpt)\n",
       "        .catch(err => showError(`Javascript Error: ${err.message}<br>This usually means there's a typo in your chart specification. See the javascript console for the full traceback.`));\n",
       "    }\n",
       "\n",
       "    if(typeof define === \"function\" && define.amd) {\n",
       "      requirejs.config({paths});\n",
       "      require([\"vega-embed\"], displayChart, err => showError(`Error loading script: ${err.message}`));\n",
       "    } else {\n",
       "      maybeLoadScript(\"vega\", \"5\")\n",
       "        .then(() => maybeLoadScript(\"vega-lite\", \"5.16.3\"))\n",
       "        .then(() => maybeLoadScript(\"vega-embed\", \"6\"))\n",
       "        .catch(showError)\n",
       "        .then(() => displayChart(vegaEmbed));\n",
       "    }\n",
       "  })({\"config\": {\"view\": {\"continuousWidth\": 300, \"continuousHeight\": 300}}, \"data\": {\"url\": \"altair-temp-data/-14656ade87b5d77bf96f3e0cac0d9578.json\", \"format\": {\"type\": \"json\"}}, \"mark\": {\"type\": \"bar\"}, \"encoding\": {\"x\": {\"bin\": {\"maxbins\": 100}, \"field\": \"value\", \"type\": \"quantitative\"}, \"y\": {\"aggregate\": \"count\", \"type\": \"quantitative\"}}, \"title\": \"Normal Distribution histogram with mean -10 and standard deviation 5\", \"$schema\": \"https://vega.github.io/schema/vega-lite/v5.16.3.json\"}, {\"mode\": \"vega-lite\"});\n",
       "</script>"
      ],
      "text/plain": [
       "alt.Chart(...)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# (1)\n",
    "import pandas as pd\n",
    "from numpy.random import default_rng\n",
    "import altair as alt\n",
    "alt.renderers.enable('default')\n",
    "\n",
    "mean_std_list = [(1,7), (10,1), (-10,5)]\n",
    "rng = default_rng(2024)\n",
    "\n",
    "\n",
    "for vals in mean_std_list:\n",
    "    norm_dist_func = stats.norm(vals[0], vals[1])\n",
    "    results = rng.normal(vals[0], vals[1], 1000)\n",
    "    normal_values = pd.DataFrame({\"value\": results})\n",
    "    norm_plot = alt.Chart(normal_values, title=\"Normal Distribution histogram with mean \" + str(vals[0]) + \" and standard deviation \" + str(vals[1])).mark_bar().encode(\n",
    "    alt.X(\"value\", bin=alt.Bin(maxbins=100)),\n",
    "    y='count()',)\n",
    "    norm_plot.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<style>\n",
       "  #altair-viz-01897036578b4edb93936d9c51cb008a.vega-embed {\n",
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       "    display: flex;\n",
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       "\n",
       "  #altair-viz-01897036578b4edb93936d9c51cb008a.vega-embed details,\n",
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       "  }\n",
       "</style>\n",
       "<div id=\"altair-viz-01897036578b4edb93936d9c51cb008a\"></div>\n",
       "<script type=\"text/javascript\">\n",
       "  var VEGA_DEBUG = (typeof VEGA_DEBUG == \"undefined\") ? {} : VEGA_DEBUG;\n",
       "  (function(spec, embedOpt){\n",
       "    let outputDiv = document.currentScript.previousElementSibling;\n",
       "    if (outputDiv.id !== \"altair-viz-01897036578b4edb93936d9c51cb008a\") {\n",
       "      outputDiv = document.getElementById(\"altair-viz-01897036578b4edb93936d9c51cb008a\");\n",
       "    }\n",
       "    const paths = {\n",
       "      \"vega\": \"https://cdn.jsdelivr.net/npm/vega@5?noext\",\n",
       "      \"vega-lib\": \"https://cdn.jsdelivr.net/npm/vega-lib?noext\",\n",
       "      \"vega-lite\": \"https://cdn.jsdelivr.net/npm/vega-lite@5.16.3?noext\",\n",
       "      \"vega-embed\": \"https://cdn.jsdelivr.net/npm/vega-embed@6?noext\",\n",
       "    };\n",
       "\n",
       "    function maybeLoadScript(lib, version) {\n",
       "      var key = `${lib.replace(\"-\", \"\")}_version`;\n",
       "      return (VEGA_DEBUG[key] == version) ?\n",
       "        Promise.resolve(paths[lib]) :\n",
       "        new Promise(function(resolve, reject) {\n",
       "          var s = document.createElement('script');\n",
       "          document.getElementsByTagName(\"head\")[0].appendChild(s);\n",
       "          s.async = true;\n",
       "          s.onload = () => {\n",
       "            VEGA_DEBUG[key] = version;\n",
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       "          };\n",
       "          s.onerror = () => reject(`Error loading script: ${paths[lib]}`);\n",
       "          s.src = paths[lib];\n",
       "        });\n",
       "    }\n",
       "\n",
       "    function showError(err) {\n",
       "      outputDiv.innerHTML = `<div class=\"error\" style=\"color:red;\">${err}</div>`;\n",
       "      throw err;\n",
       "    }\n",
       "\n",
       "    function displayChart(vegaEmbed) {\n",
       "      vegaEmbed(outputDiv, spec, embedOpt)\n",
       "        .catch(err => showError(`Javascript Error: ${err.message}<br>This usually means there's a typo in your chart specification. See the javascript console for the full traceback.`));\n",
       "    }\n",
       "\n",
       "    if(typeof define === \"function\" && define.amd) {\n",
       "      requirejs.config({paths});\n",
       "      require([\"vega-embed\"], displayChart, err => showError(`Error loading script: ${err.message}`));\n",
       "    } else {\n",
       "      maybeLoadScript(\"vega\", \"5\")\n",
       "        .then(() => maybeLoadScript(\"vega-lite\", \"5.16.3\"))\n",
       "        .then(() => maybeLoadScript(\"vega-embed\", \"6\"))\n",
       "        .catch(showError)\n",
       "        .then(() => displayChart(vegaEmbed));\n",
       "    }\n",
       "  })({\"config\": {\"view\": {\"continuousWidth\": 300, \"continuousHeight\": 300}}, \"data\": {\"url\": \"altair-temp-data/-a484195b8908187c70a43751fa536ef8.json\", \"format\": {\"type\": \"json\"}}, \"mark\": {\"type\": \"bar\"}, \"encoding\": {\"x\": {\"bin\": {\"maxbins\": 100}, \"field\": \"value\", \"type\": \"quantitative\"}, \"y\": {\"aggregate\": \"count\", \"type\": \"quantitative\"}}, \"title\": \"Normal Distribution histogram with mean 1 and standard deviation 2\", \"$schema\": \"https://vega.github.io/schema/vega-lite/v5.16.3.json\"}, {\"mode\": \"vega-lite\"});\n",
       "</script>"
      ],
      "text/plain": [
       "alt.Chart(...)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# (2)\n",
    "def display_norm(mean, std):\n",
    "    norm_dist_func = stats.norm(mean, std)\n",
    "    results = norm_dist_func.rvs(1000)\n",
    "    normal_values = pd.DataFrame({\"value\": results})\n",
    "    norm_plot = alt.Chart(normal_values, title=\"Normal Distribution histogram with mean \" + str(mean) + \" and standard deviation \" + str(std)).mark_bar().encode(\n",
    "    alt.X(\"value\", bin=alt.Bin(maxbins=100)),\n",
    "    y='count()',)\n",
    "    norm_plot.display()\n",
    "    \n",
    "display_norm(1,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<style>\n",
       "  #altair-viz-7e742cb62c564150b5f8738d36227edb.vega-embed {\n",
       "    width: 100%;\n",
       "    display: flex;\n",
       "  }\n",
       "\n",
       "  #altair-viz-7e742cb62c564150b5f8738d36227edb.vega-embed details,\n",
       "  #altair-viz-7e742cb62c564150b5f8738d36227edb.vega-embed details summary {\n",
       "    position: relative;\n",
       "  }\n",
       "</style>\n",
       "<div id=\"altair-viz-7e742cb62c564150b5f8738d36227edb\"></div>\n",
       "<script type=\"text/javascript\">\n",
       "  var VEGA_DEBUG = (typeof VEGA_DEBUG == \"undefined\") ? {} : VEGA_DEBUG;\n",
       "  (function(spec, embedOpt){\n",
       "    let outputDiv = document.currentScript.previousElementSibling;\n",
       "    if (outputDiv.id !== \"altair-viz-7e742cb62c564150b5f8738d36227edb\") {\n",
       "      outputDiv = document.getElementById(\"altair-viz-7e742cb62c564150b5f8738d36227edb\");\n",
       "    }\n",
       "    const paths = {\n",
       "      \"vega\": \"https://cdn.jsdelivr.net/npm/vega@5?noext\",\n",
       "      \"vega-lib\": \"https://cdn.jsdelivr.net/npm/vega-lib?noext\",\n",
       "      \"vega-lite\": \"https://cdn.jsdelivr.net/npm/vega-lite@5.16.3?noext\",\n",
       "      \"vega-embed\": \"https://cdn.jsdelivr.net/npm/vega-embed@6?noext\",\n",
       "    };\n",
       "\n",
       "    function maybeLoadScript(lib, version) {\n",
       "      var key = `${lib.replace(\"-\", \"\")}_version`;\n",
       "      return (VEGA_DEBUG[key] == version) ?\n",
       "        Promise.resolve(paths[lib]) :\n",
       "        new Promise(function(resolve, reject) {\n",
       "          var s = document.createElement('script');\n",
       "          document.getElementsByTagName(\"head\")[0].appendChild(s);\n",
       "          s.async = true;\n",
       "          s.onload = () => {\n",
       "            VEGA_DEBUG[key] = version;\n",
       "            return resolve(paths[lib]);\n",
       "          };\n",
       "          s.onerror = () => reject(`Error loading script: ${paths[lib]}`);\n",
       "          s.src = paths[lib];\n",
       "        });\n",
       "    }\n",
       "\n",
       "    function showError(err) {\n",
       "      outputDiv.innerHTML = `<div class=\"error\" style=\"color:red;\">${err}</div>`;\n",
       "      throw err;\n",
       "    }\n",
       "\n",
       "    function displayChart(vegaEmbed) {\n",
       "      vegaEmbed(outputDiv, spec, embedOpt)\n",
       "        .catch(err => showError(`Javascript Error: ${err.message}<br>This usually means there's a typo in your chart specification. See the javascript console for the full traceback.`));\n",
       "    }\n",
       "\n",
       "    if(typeof define === \"function\" && define.amd) {\n",
       "      requirejs.config({paths});\n",
       "      require([\"vega-embed\"], displayChart, err => showError(`Error loading script: ${err.message}`));\n",
       "    } else {\n",
       "      maybeLoadScript(\"vega\", \"5\")\n",
       "        .then(() => maybeLoadScript(\"vega-lite\", \"5.16.3\"))\n",
       "        .then(() => maybeLoadScript(\"vega-embed\", \"6\"))\n",
       "        .catch(showError)\n",
       "        .then(() => displayChart(vegaEmbed));\n",
       "    }\n",
       "  })({\"config\": {\"view\": {\"continuousWidth\": 300, \"continuousHeight\": 300}}, \"data\": {\"url\": \"altair-temp-data/-ba10808476d179cf9cfe9862afe696fe.json\", \"format\": {\"type\": \"json\"}}, \"mark\": {\"type\": \"area\", \"interpolate\": \"step\", \"opacity\": 0.8}, \"encoding\": {\"color\": {\"field\": \"Columns\", \"type\": \"nominal\"}, \"x\": {\"bin\": {\"maxbins\": 300}, \"field\": \"Values\", \"type\": \"quantitative\"}, \"y\": {\"aggregate\": \"count\", \"stack\": null, \"type\": \"quantitative\"}}, \"transform\": [{\"fold\": [\"Norm 1\", \"Norm 3\"], \"as\": [\"Columns\", \"Values\"]}], \"$schema\": \"https://vega.github.io/schema/vega-lite/v5.16.3.json\"}, {\"mode\": \"vega-lite\"});\n",
       "</script>"
      ],
      "text/plain": [
       "alt.Chart(...)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# (3) Compound Histogram Altair\n",
    "df = pd.DataFrame({'Norm 1': stats.norm(1,7).rvs(6000),\n",
    "                   'Norm 2': stats.norm(10,1).rvs(6000),\n",
    "                   'Norm 3': stats.norm(-10,5).rvs(6000)})\n",
    "\n",
    "alt.Chart(df).transform_fold(\n",
    "    ['Norm 1', 'Norm 3'],\n",
    "    as_=['Columns', 'Values']\n",
    ").mark_area(\n",
    "    opacity=0.8,\n",
    "    interpolate='step'\n",
    ").encode(\n",
    "    alt.X('Values:Q', bin=alt.Bin(maxbins=300)),\n",
    "    alt.Y('count()', stack=None),\n",
    "    alt.Color('Columns:N'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*For solutions, see `solutions/normal_distributions.py`*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Probability Density Functions\n",
    "\n",
    "The histograms we were looking at above are approximations of what is known as a *Probability Density Function*, or PDF. A PDF shows us the probability of a given value, for a discrete random variable.\n",
    "\n",
    "For a continuous random variable, any *exact* value has a probability of 0 (think about why this is the case). We can only give probabilities for ranges. For a lower bound of $a$ and an upper bound of $b$, we can obtain the probability of a random value lying between $a$ and $b$ by integrating the PDF between those values. For example, if our PDF function is $f(x)$, then the probability of random variable $X$ lying between $a$ and $b$ is given by:\n",
    "\n",
    "$P[a <= X <= b] = \\int_a^b f(x) dx$\n",
    "\n",
    "As an aside, and very informally, you can think of \"the probability of $X=2$\" as being a range *about* 2, say any value between 1.99 and 2.01 (or whatever precision makes sense in the context). Be clear on this definition when you do go to present your findings though, as a formal definition will be needed to replicate your results.\n",
    "\n",
    "We can generate the PDF through the `.pdf` method on a `scipy.stats` distribution. Earlier in this notebook we computed this manually through the equation. Now we let a library do the work for us."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = stats.norm(0, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<scipy.stats._distn_infrastructure.rv_continuous_frozen at 0x17d59a66510>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.linspace(-4, 4, 1000)\n",
    "y = n.pdf(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As with all probabilities, the sum or integral of the probabilities of all possible outcomes must equal one. In a die roll, each outcome (1, 2, 3, 4, 5, 6) has an probability of $\\frac{1}{6}$. Summed, that equals one.\n",
    "\n",
    "In a continuous scenario, the *area under the curve* must sum to 1.0 across the full range. That is:\n",
    "\n",
    "$\\int_{-\\infty}^{\\infty}f(x)dx = 1$\n",
    "\n",
    "<div class=\"alert alert-warning\">\n",
    "    Note that the value $f(x)$ is not a probability! It might look like one, but it's the <i>area under the curve</i> that gives us the probability.\n",
    "</div>\n",
    "\n",
    "For a given function, if it is positive or 0 everywhere, and the area under the curve integrates to 1, the function is a valid probability density function.\n",
    "\n",
    "Some functions are valid only under a given domain. For instance, the function $y = ax(b-x)$ is only positive for $0<=x<=b$. In this case, if the area under the curve sums to 1 across that domain, the function is a valid PDF for that domain.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "#### Exercises\n",
    "Using three different distribution classes in `scipy.stats`, run the following experiment:\n",
    "\n",
    "1. Compute 10,000 random values. Compute and save the mean.\n",
    "2. Repeat step 1, 1,000 times\n",
    "3. Plot the histogram of the mean values and observe the shape.\n",
    "\n",
    "Use the distributions `norm`, `cosine` and `uniform`.\n",
    "\n",
    "This finding is why the normal distribution is so important, and so commonly found in nature.\n",
    "\n",
    "**Hint:** See this example for how to do independent axes with Altair: https://altair-viz.github.io/gallery/layered_plot_with_dual_axis.html?highlight=resolve_scale\n",
    "\n",
    "#### Extended exercise\n",
    "\n",
    "Load a random dataset, and perform the same exercise with the real world data - that is, take the mean of each of 1,000 samples of size 10,000 and plot the histogram. What shaped distribution does it look like?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
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       "          s.onload = () => {\n",
       "            VEGA_DEBUG[key] = version;\n",
       "            return resolve(paths[lib]);\n",
       "          };\n",
       "          s.onerror = () => reject(`Error loading script: ${paths[lib]}`);\n",
       "          s.src = paths[lib];\n",
       "        });\n",
       "    }\n",
       "\n",
       "    function showError(err) {\n",
       "      outputDiv.innerHTML = `<div class=\"error\" style=\"color:red;\">${err}</div>`;\n",
       "      throw err;\n",
       "    }\n",
       "\n",
       "    function displayChart(vegaEmbed) {\n",
       "      vegaEmbed(outputDiv, spec, embedOpt)\n",
       "        .catch(err => showError(`Javascript Error: ${err.message}<br>This usually means there's a typo in your chart specification. See the javascript console for the full traceback.`));\n",
       "    }\n",
       "\n",
       "    if(typeof define === \"function\" && define.amd) {\n",
       "      requirejs.config({paths});\n",
       "      require([\"vega-embed\"], displayChart, err => showError(`Error loading script: ${err.message}`));\n",
       "    } else {\n",
       "      maybeLoadScript(\"vega\", \"5\")\n",
       "        .then(() => maybeLoadScript(\"vega-lite\", \"5.16.3\"))\n",
       "        .then(() => maybeLoadScript(\"vega-embed\", \"6\"))\n",
       "        .catch(showError)\n",
       "        .then(() => displayChart(vegaEmbed));\n",
       "    }\n",
       "  })({\"config\": {\"view\": {\"continuousWidth\": 300, \"continuousHeight\": 300}}, \"data\": {\"url\": \"altair-temp-data/-686717efed30c23dd4d3c5b00493cb57.json\", \"format\": {\"type\": \"json\"}}, \"mark\": {\"type\": \"bar\"}, \"encoding\": {\"x\": {\"bin\": {\"maxbins\": 100}, \"field\": \"normal\", \"type\": \"quantitative\"}, \"y\": {\"aggregate\": \"count\", \"type\": \"quantitative\"}}, \"$schema\": \"https://vega.github.io/schema/vega-lite/v5.16.3.json\"}, {\"mode\": \"vega-lite\"});\n",
       "</script>"
      ],
      "text/plain": [
       "alt.Chart(...)"
      ]
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     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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       "\n",
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       "  #altair-viz-fbeb9ca7fa1d4c1281895ca922da47ee.vega-embed {\n",
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       "    display: flex;\n",
       "  }\n",
       "\n",
       "  #altair-viz-fbeb9ca7fa1d4c1281895ca922da47ee.vega-embed details,\n",
       "  #altair-viz-fbeb9ca7fa1d4c1281895ca922da47ee.vega-embed details summary {\n",
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       "  }\n",
       "</style>\n",
       "<div id=\"altair-viz-fbeb9ca7fa1d4c1281895ca922da47ee\"></div>\n",
       "<script type=\"text/javascript\">\n",
       "  var VEGA_DEBUG = (typeof VEGA_DEBUG == \"undefined\") ? {} : VEGA_DEBUG;\n",
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       "      outputDiv = document.getElementById(\"altair-viz-fbeb9ca7fa1d4c1281895ca922da47ee\");\n",
       "    }\n",
       "    const paths = {\n",
       "      \"vega\": \"https://cdn.jsdelivr.net/npm/vega@5?noext\",\n",
       "      \"vega-lib\": \"https://cdn.jsdelivr.net/npm/vega-lib?noext\",\n",
       "      \"vega-lite\": \"https://cdn.jsdelivr.net/npm/vega-lite@5.16.3?noext\",\n",
       "      \"vega-embed\": \"https://cdn.jsdelivr.net/npm/vega-embed@6?noext\",\n",
       "    };\n",
       "\n",
       "    function maybeLoadScript(lib, version) {\n",
       "      var key = `${lib.replace(\"-\", \"\")}_version`;\n",
       "      return (VEGA_DEBUG[key] == version) ?\n",
       "        Promise.resolve(paths[lib]) :\n",
       "        new Promise(function(resolve, reject) {\n",
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       "          document.getElementsByTagName(\"head\")[0].appendChild(s);\n",
       "          s.async = true;\n",
       "          s.onload = () => {\n",
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       "          };\n",
       "          s.onerror = () => reject(`Error loading script: ${paths[lib]}`);\n",
       "          s.src = paths[lib];\n",
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       "\n",
       "    function showError(err) {\n",
       "      outputDiv.innerHTML = `<div class=\"error\" style=\"color:red;\">${err}</div>`;\n",
       "      throw err;\n",
       "    }\n",
       "\n",
       "    function displayChart(vegaEmbed) {\n",
       "      vegaEmbed(outputDiv, spec, embedOpt)\n",
       "        .catch(err => showError(`Javascript Error: ${err.message}<br>This usually means there's a typo in your chart specification. See the javascript console for the full traceback.`));\n",
       "    }\n",
       "\n",
       "    if(typeof define === \"function\" && define.amd) {\n",
       "      requirejs.config({paths});\n",
       "      require([\"vega-embed\"], displayChart, err => showError(`Error loading script: ${err.message}`));\n",
       "    } else {\n",
       "      maybeLoadScript(\"vega\", \"5\")\n",
       "        .then(() => maybeLoadScript(\"vega-lite\", \"5.16.3\"))\n",
       "        .then(() => maybeLoadScript(\"vega-embed\", \"6\"))\n",
       "        .catch(showError)\n",
       "        .then(() => displayChart(vegaEmbed));\n",
       "    }\n",
       "  })({\"config\": {\"view\": {\"continuousWidth\": 300, \"continuousHeight\": 300}}, \"data\": {\"url\": \"altair-temp-data/-d4623a37fe15f94917ad806d9664a33c.json\", \"format\": {\"type\": \"json\"}}, \"mark\": {\"type\": \"bar\"}, \"encoding\": {\"x\": {\"bin\": {\"maxbins\": 100}, \"field\": \"poisson\", \"type\": \"quantitative\"}, \"y\": {\"aggregate\": \"count\", \"type\": \"quantitative\"}}, \"$schema\": \"https://vega.github.io/schema/vega-lite/v5.16.3.json\"}, {\"mode\": \"vega-lite\"});\n",
       "</script>"
      ],
      "text/plain": [
       "alt.Chart(...)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "\n",
       "<style>\n",
       "  #altair-viz-6488fbb47e0345de9ea8144107f2d6ef.vega-embed {\n",
       "    width: 100%;\n",
       "    display: flex;\n",
       "  }\n",
       "\n",
       "  #altair-viz-6488fbb47e0345de9ea8144107f2d6ef.vega-embed details,\n",
       "  #altair-viz-6488fbb47e0345de9ea8144107f2d6ef.vega-embed details summary {\n",
       "    position: relative;\n",
       "  }\n",
       "</style>\n",
       "<div id=\"altair-viz-6488fbb47e0345de9ea8144107f2d6ef\"></div>\n",
       "<script type=\"text/javascript\">\n",
       "  var VEGA_DEBUG = (typeof VEGA_DEBUG == \"undefined\") ? {} : VEGA_DEBUG;\n",
       "  (function(spec, embedOpt){\n",
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       "    if (outputDiv.id !== \"altair-viz-6488fbb47e0345de9ea8144107f2d6ef\") {\n",
       "      outputDiv = document.getElementById(\"altair-viz-6488fbb47e0345de9ea8144107f2d6ef\");\n",
       "    }\n",
       "    const paths = {\n",
       "      \"vega\": \"https://cdn.jsdelivr.net/npm/vega@5?noext\",\n",
       "      \"vega-lib\": \"https://cdn.jsdelivr.net/npm/vega-lib?noext\",\n",
       "      \"vega-lite\": \"https://cdn.jsdelivr.net/npm/vega-lite@5.16.3?noext\",\n",
       "      \"vega-embed\": \"https://cdn.jsdelivr.net/npm/vega-embed@6?noext\",\n",
       "    };\n",
       "\n",
       "    function maybeLoadScript(lib, version) {\n",
       "      var key = `${lib.replace(\"-\", \"\")}_version`;\n",
       "      return (VEGA_DEBUG[key] == version) ?\n",
       "        Promise.resolve(paths[lib]) :\n",
       "        new Promise(function(resolve, reject) {\n",
       "          var s = document.createElement('script');\n",
       "          document.getElementsByTagName(\"head\")[0].appendChild(s);\n",
       "          s.async = true;\n",
       "          s.onload = () => {\n",
       "            VEGA_DEBUG[key] = version;\n",
       "            return resolve(paths[lib]);\n",
       "          };\n",
       "          s.onerror = () => reject(`Error loading script: ${paths[lib]}`);\n",
       "          s.src = paths[lib];\n",
       "        });\n",
       "    }\n",
       "\n",
       "    function showError(err) {\n",
       "      outputDiv.innerHTML = `<div class=\"error\" style=\"color:red;\">${err}</div>`;\n",
       "      throw err;\n",
       "    }\n",
       "\n",
       "    function displayChart(vegaEmbed) {\n",
       "      vegaEmbed(outputDiv, spec, embedOpt)\n",
       "        .catch(err => showError(`Javascript Error: ${err.message}<br>This usually means there's a typo in your chart specification. See the javascript console for the full traceback.`));\n",
       "    }\n",
       "\n",
       "    if(typeof define === \"function\" && define.amd) {\n",
       "      requirejs.config({paths});\n",
       "      require([\"vega-embed\"], displayChart, err => showError(`Error loading script: ${err.message}`));\n",
       "    } else {\n",
       "      maybeLoadScript(\"vega\", \"5\")\n",
       "        .then(() => maybeLoadScript(\"vega-lite\", \"5.16.3\"))\n",
       "        .then(() => maybeLoadScript(\"vega-embed\", \"6\"))\n",
       "        .catch(showError)\n",
       "        .then(() => displayChart(vegaEmbed));\n",
       "    }\n",
       "  })({\"config\": {\"view\": {\"continuousWidth\": 300, \"continuousHeight\": 300}}, \"data\": {\"url\": \"altair-temp-data/-4908a54084a15ea408fed36370102733.json\", \"format\": {\"type\": \"json\"}}, \"mark\": {\"type\": \"bar\"}, \"encoding\": {\"x\": {\"bin\": {\"maxbins\": 100}, \"field\": \"uniform\", \"type\": \"quantitative\"}, \"y\": {\"aggregate\": \"count\", \"type\": \"quantitative\"}}, \"$schema\": \"https://vega.github.io/schema/vega-lite/v5.16.3.json\"}, {\"mode\": \"vega-lite\"});\n",
       "</script>"
      ],
      "text/plain": [
       "alt.Chart(...)"
      ]
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     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# (1)\n",
    "\n",
    "def get_rv_sample_mean(distribution, n_sample = 10000):\n",
    "    samp = distribution.rvs(n_sample)\n",
    "    return samp.mean()\n",
    "\n",
    "\n",
    "def list_of_means(distribution, n_means = 1000, n_sample = 10000):\n",
    "    samp_means = []\n",
    "    for i in range(n_means):\n",
    "        samp_means.append(get_rv_sample_mean(distribution, n_sample))\n",
    "    return samp_means\n",
    "\n",
    "\n",
    "distributions = dict([(\"normal\", stats.norm(0,1)),\n",
    "                 (\"poisson\", stats.poisson(1,1)),\n",
    "                 (\"uniform\", stats.uniform(0, 1))\n",
    "                ])\n",
    "\n",
    "\n",
    "means = pd.DataFrame({distribution_name: list_of_means(distribution)\n",
    "                  for distribution_name, distribution in distributions.items()})\n",
    "\n",
    "\n",
    "for dist in distributions:\n",
    "    nplot = alt.Chart(means[[dist]]).mark_bar().encode(alt.X(dist, bin=alt.Bin(maxbins=100)), y='count()',)\n",
    "    nplot.display()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*For solutions, see `solutions/plotting_distributions.py`*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cumulative Density Function\n",
    "\n",
    "A cumulative density function (CDF) tells us the probability of a value being less than, or equal to, a given value. A CDF is normally more useful for everyday usage.\n",
    "\n",
    "Here is the CDF for the normal distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "distribution = stats.norm()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.linspace(-5, 5, 1000)\n",
    "y_cdf = distribution.cdf(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y_cdf);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "This graph allows us to compute the probability that a random value is chosen from the normal distribution ($\\mu=0, \\sigma=1$) will be less than or equal to the given $x$ value.\n",
    "\n",
    "For example, to determine the probability a given value is less than or equal to 1.2, we find the $y$ value corresponding to the given $x$ value:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "At x=1.2, y=0.885\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_0 = 1.2\n",
    "y_0 = distribution.cdf(x_0)\n",
    "\n",
    "plt.plot(x, y_cdf)\n",
    "plt.plot([x[0], x_0], [y_0, y_0], '--')\n",
    "plt.plot([x_0, x_0], [0, y_0], '--')\n",
    "print(\"At x={0}, y={1:.3f}\".format(x_0, y_0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some major properties of a CDF, defined as $F_X(x)$ are:\n",
    " \n",
    "* The range (the values y can take) are between (and including) 0 and 1. This comes from its use as a probability computation - see the next point. Formally, $0 <= F_X(x) <= 1$ for any value $x$\n",
    "* The likelihood a value is less than or equal to infinity must be 1, and the likelihood it is less than or equal to negative infinity must be 0. Formally, $F_X(-\\infty)=0$ and $F_X(\\infty)=1$\n",
    "* The CDF value is *increasing*. That is, it must have a positive or zero slope (but not a negative one). Formally, $F_X(a) <= F_X(b) \\text{ iff } a < b$\n",
    " \n",
    "Further, if your random variable has a domain, then the CDF is 0 for all values less than the minimum domain, and 1 for any value above the maximum of the domain.\n",
    "\n",
    "To find the probability that a random variable lies within a range, subtract the value of the CDF at the minimum from the value at the maximum:\n",
    "\n",
    "$P(a < X <= b) = F_X(b) - F_X(a)$\n",
    "\n",
    "For a continuous distribution (like the normal distribution), the CDF is a smooth graph. Its shape will vary, but will follow the general pattern as seen above.\n",
    "\n",
    "For a discrete distribution, the graph is still increasing, but represents a \"staircase\", where the value of $F_X(x)$ between valid values remains the same. For example, for a die roll, the value of $F_X(3) == F_X(3.1)$. This makes sense, as no values between 3.0 and 3.1 can be obtained on a die, so the value $P(X<=3)$ must be the same as $P(X<=3.1)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercises\n",
    "\n",
    "1. Plot the relationship between the mean of a normal distribution as $x$ (with a fixed standard deviation of 1), and the area under the curve between 0 and 1 as the $y$ axis.\n",
    "2. Remake the plot, but have the standard deviation vary as $x$, and the mean fixed as 0. How can you characterise the relationship?\n",
    "\n",
    "#### Extended exercise\n",
    "\n",
    "Plot the CDF for rolling two dice, where the x value is the sum of the two dice, and the y value is the CDF for the distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The area between 0 and 1 decreases as the standard deviation increases are values will spread out and hence have less between 0 and 1.\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Exercise 1\n",
    "x_values_1 = np.linspace(-5, 5, 1000)\n",
    "distributions_1 = [stats.norm(x, 1) for x in x_values_1]\n",
    "y_1 = np.array([distribution.cdf(1) - distribution.cdf(0)\n",
    "               for distribution in distributions_1])\n",
    "plt.plot(x_values_1, y_1, \"r\")\n",
    "\n",
    "# Exercise 2\n",
    "x_values_2 = np.linspace(0.1, 20, 1000)\n",
    "distributions_2 = [stats.norm(0, x) for x in x_values_2]\n",
    "y_2 = np.array([distribution.cdf(1) - distribution.cdf(0)\n",
    "               for distribution in distributions_2])\n",
    "plt.plot(x_values_2, y_2)\n",
    "print(\"The area between 0 and 1 decreases as the standard deviation increases are values will spread out and hence have less between 0 and 1.\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*For solutions, see `solutions/cdf_relationships.py`*"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}
